Don't see a clair text in basic concepts of Mathematical Logic. This one is not pretend to be at the begining. This saved some complecities that otherwise didn't doing better....

Simiple statement .....

Compound statement Finite composition of finite number
of simple statements and connectives $\lnot,\;\wedge,\,\vee,\,\to,\,\leftrightarrow.$
In other words, we call statements of the form
$\qquad A,\,\lnot A,\, A\wedge B,\,A\vee B,\,A\to B,\, A\leftrightarrow B\qquad(\dagger)$
compound statements where $A,\,B$ are simple statements.
Now inductively statements of the form $\,(\dagger)\,$are called
compound statements if$\,A,\,B\,$are compound statements.

I always wondering what a veriable mean, or how it be defined in set theory term.

I guess is a veriable is a pair $(v,\mathbb{id}_D)$ where $\mathbb{id}_D: u\mapsto u$ is the identity mapping on a set $D$ while $v$ is a lable or a name of the mapping. Usually, for short we use $v$ instead of $(v,\mathbb{id}_D)$